Let's that, let that be spatial frequency in the x direction.

Let's, let that be spatial frequency in the z direction and

then let's let spatial frequency in the y direction be into the plane.

Now let's suppose the very first image that we recorded as

a projection of an untilted sample, then the amplitudes and

phases in the transform of that image will be the amplitudes and

phases on the special frequency in x and y planes,

then we tilt the sample just a little bit and record another image.

And so these will be the amplitudes and phases of say,

the first tilted image and then we tilt the image again.

We tilt the sample again and record a second image and

its Fourier transform fills reciprocal space there.

And as you can see, as we record images to higher and higher tilt angle,

we will sample, measure the amplitudes and phases in reciprocal space.

Now, in tomography, we typically can only get up to about 60,

65 maybe even 70 degrees tilt before the sample itself becomes too thick.

Of course, at 90 degrees, it would be infinitely thick and so

we can only record useful images up to 60 or 70 degrees.

And so there's a missing region here of data that we don't record.

Now let's suppose we started tilting our sample in the opposite direction and

recorded images towards what you might call negative tilt angles and

then we would sample these amplitudes and phases in the negative direction.

And again, we couldn't tilt past about minus negative 60 or negative 70 degrees.

Now, each of the images that we recorded is digital of course.

And so when we calculate the Fourier transform, it's a series of equally spaces

pixels, each pixel contains an amplitude and a phase.

And so we might represent that by little circles representing each

measurement of an amplitude and phase.

And the first image gives us these measurements and

I'll try to make them equally spaced.

Okay.

And likewise, on the first tilted image,

we get evenly spaced measurements of amplitude and phase along that plane.

And so we get a series of measurements like this.

And of course, here as well.

And the second image gives us equally spaced

measurements of amplitude and face and so forth.

But in order to do an inverse 3D Fourier transform to generate

the three-dimensional reconstruction,

what we need are the amplitudes and phases on a regular lattice.

For instance, a Cartesian coordinate system.

And so we need amplitudes in phases on the intersections of all of these points and

so we have an interpolation problem.

So for instance, if we need to find the amplitude in

phase of the 3D Fourier transform right in that position.

We have at our disposal a number of measurements in the vicinity and

so there's different algorithms to merge the amplitudes and phases.

More sophisticated algorithms use amplitudes and phases further and

further away.

But at the end of the day, they all make an estimate of what is the amplitude and

phase of the 3D Fourier transform at a regular lattice point.

And so we make these estimates as we go out into further and

further into reciprocal space and then we calculate an inverse

3D Fourier transform to generate the reconstruction of our object.

So, in the case of the tilt series that I showed you as an example of

the Bdellovibrio bacterial cell,

this is what the three-dimensional reconstruction looks like.

And here in this movie, I'm showing it from the bottom of the cell up to

the top of the cell from the perspective of the electron beam.

And in the movie, it's shown slice by slice.

And so by tomography,

we've been able to resolve the interior features of that cell in 3D.

And we call such a three-dimensional reconstruction from tomography,

we call it a tomogram.

And so, in order to better appreciate the information content of this tomogram of

the cell, it's hard to see things in detail when we're moving slice by

slice up and down through the cell.

Here, I'm showing just a single slice through the cell and

I'm showing it in duplicate.

These are identical copies, so that I can draw on this one and

you can still look at the other one without the drawing on it.

In this tomogram, we can clearly see an outer membrane.

We see an inner membrane of the cell, you can even see the peptidoglycan layer,

the cell wall in between them.

Do you see that there's a region here, I'll outline it,

that really has a different texture inside the cell than the surround.

These larger dark objects that are about 25 nanometers in size and they're darker.

These are ribosomes inside the cell.

They're darker, because they contain a lot of phosphorus and so

they scatter more strongly.

And we know from their size that you know,

ribosomes are about 25 nanometers in diameter.

And so do you see that in the middle of the cell here,

we have a region that has a really different texture.

There don't seem to be any ribosomes in here.

And there some swirly patterns and some linear features.

This is presumably the nuclide where the DNA inside this cell is being packed

together.

And finally, at the pole of this cell, there is a flagellum.

So here's a flagellum and at the base of the flagellum is a flagellar motor.

And to get a better appreciation of the detail present in such tomograms.

Here is an enlarged view of that single slice of the flageller motor.

Again, in duplicate, so I can draw on one, while you can still see the other one.

And do you see this density here?

And this density here, for instance?

These two densities form a cup called the C ring.

We're viewing that C ring in cross section, so

you just see this side of it and this side of it, but it's actually a whole cup.

And you see this density here and this density here, we've been able to show

that that's the cytoplasmic domain of a protein called Fla-H.

In addition, we can see protein layers here and

here there's an important protein piece.

There's some kind of a plate that exists in the periplasm here.

There are, are clear densities as part of the M, the MS ring.

And anchor points to the growing flagellum here as it emerges from the cell.

And so there is present in these cryotomograms enough resolution

to resolve,

even the shapes of protein complexes within the cell in a near native state.

Now because tomography produces three-dimensional

reconstructions of our samples, we can then interrogate it in 3D.

And this is illustrated by the following movie.

This is another bacterial cell in a tomogram being presented from the back to

the front, slice by slice, like the previous one.

But here on the second pass,

we've identified on the computer object's of interest.

Here are filament and gold and some iron core,

iron objects inside the cell called magnetosomes.

And finally, we layer on the membranes, the outer membrane and

the inner membrane of this cell.

And because it's in 3D, we can then move into the cell and

interrogate the structure from any angle that we'd like.

Here, we were interested in the relationship between the vesicles and

the inner membrane and so tomography gives you a 3D reconstruction.

Now tomography can be done on very large objects and very small objects.

Let's first talk about the very large.

Suppose that we had a large block of tissue, for instance,

that even contain many cells inside of it.

We can use a microtome to section this block of tissue and

then each section, if we were to lay it down, for

instance, we might see a pattern of cells on it.

And then the next section would have a different slice through those same cells.

And finally, the next section would have yet another slice through those cells.

And so we call this serial sectioning.

Now, on the scale of a cell,

the typical field of view of a tomogram might be very small,

might be just a small piece of one of those cells.

And in order to expand the area that we can record in a tomogram,

it's possible to do what's called montage tomography.

And for the purposes of illustration, I'm going to enlarge the,

the kind of field of view you might get in a tomogram suppose it was this large.

Well, you could record a tomogram of this region and

then you could record another tilt series of the adjacent region and

another tilt series of the adjacent region and so on.

And you'd want to have a little bit of overlap, so

that you could stitch them together later.

And then you could record another tilt series of each of these frames.

And finally, perhaps another row of frames here.

And they're always overlapped with ones next to them and above and below them, so

that you can stitch them together.

And then if we did the same thing on the next slice, et cetera.

And the next slice, et cetera, et cetera.

Then we could combine all these tomograms and

stitch them together in, in the final reconstruction.

And this strategy is called serial section montage and

then if you were to record tilt series and

produce 3D reconstruction, montage tomography.

The serial section refers to sectioning, you know,

collecting serial sections through the cell in one dimension.

The montage refers to the fact that instead of recording just a single tilt

series of a small area on that cell, we would record a whole set of tilt series,

overlapping tilt series.

And then later, stitch them together in the reconstruction process.

And this can be extended, so

you can reconstruct even very large regions of a cell.

Some of the most ambitious research projects have been to produce

three-dimensional reconstructions of entire cells for instance,

entire human cells by this serial section montage tomography.

As an example of a serial section tomogram, I'm going to show you one

through a Golgi complex that was recorded by Mark Ledinski.

And to show you the serial sections, this is a light microscope image

of five serial sections that mark cut of this object of interest.

Now the object of interest is a very small part of a cell that's right here

in the sections.

And this section wasn't imaged completely, but

these dark star shaped patterns were produced by imaging

the section in the electron microscope in a dual axis tilt series.

So, one axis tilt series produced one of these elliptical burn marks and

the other axis produced this other elliptical burn mark.

And so mark image this section and then the imaged the same feature of the cell

in the next section and the next section.

And then finally, a fourth section.

Now, after three dimensional reconstructions were calculated from

each of those dual axis tilt series, those four reconstructions were

then stitched together to produce a much larger reconstruction of a bigger

volume of the Golgi and it looks like this form the bottom to the top.

So here, we're going slice by slice through all four of those sections and

between the sections, you see a little blurriness,

because the stitching doesn't work exactly right.

But nevertheless, you can see certain Golgi cisternae progress

all the way from the bottom to the top.

There is incredible amount of detail in these reconstructions.

You can follow all the membranes and the vesicles and the Golgi cisternae.

Now, once that's done, typically we segment the objects of interest.

And so here each of the cisternae and vesicles involved in the Golgi are colored

in different colors, so that we can then dissect them later.

Look at them individually, if we like.

Look at the interrelationship between them.